CN114890305A - Self-adaptive backstepping nonlinear swing suppression control method of double-pendulum rotary crane - Google Patents
Self-adaptive backstepping nonlinear swing suppression control method of double-pendulum rotary crane Download PDFInfo
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Abstract
The invention discloses a self-adaptive backstepping nonlinear swing suppression control method of a double-pendulum rotary crane, which comprises the following steps: establishing a mathematical model of the double-pendulum rotary crane based on a Lagrange kinetic equation and analyzing characteristics; establishing an energy function of a mathematical model of the rotary crane according to the characteristics, establishing a Lyapunov function based on a backstepping method and a system control target, and establishing an adaptive controller based on the energy function and the Lyapunov function; and obtaining the self-adaptive control rate based on the Lyapunov function, and adding the self-adaptive control rate into the self-adaptive controller. The design of the self-adaptive control rate has good robustness on system parameters of different rotary cranes; the control method of the PD-like in the self-adaptive controller can realize accurate positioning, and external disturbance is restrained through the feedback of the swing angle information and the design of an error item; the two are combined to realize efficient track tracking and swing suppression of different crane systems, so that a control effect is achieved.
Description
Technical Field
The invention relates to the technical field of motion control of an under-actuated crane system, in particular to a self-adaptive backstepping nonlinear swing suppression control method of a double-pendulum rotary crane.
Background
Under-actuated systems, i.e. systems where the system inputs fewer degrees of freedom than the system. Among these, crane systems are typical under-actuated systems, where the angle of the load and hook is such that it cannot directly control the force. And the device has the advantages of simple structure, low power consumption, few actuating mechanisms, wide application occasions and the like. Generally, the conventional swing eliminating method of a crane is mostly operated by people, but this causes many problems because people's operation is different from machine operation, and accurate positioning and swing elimination cannot be realized. The imprecise operation of which can cause errors and can even be dangerous.
Disclosure of Invention
This section is for the purpose of summarizing some aspects of embodiments of the invention and to briefly introduce some preferred embodiments. In this section, as well as in the abstract and the title of the invention of this application, simplifications or omissions may be made to avoid obscuring the purpose of the section, the abstract and the title, and such simplifications or omissions are not intended to limit the scope of the invention.
The present invention has been made in view of the above-mentioned problems.
Therefore, the technical problem solved by the invention is as follows: under the conditions that fluctuation and rotation actions of a cantilever of the double-pendulum rotary crane influence the swinging of a load and the load cannot be directly controlled, how to simultaneously realize accurate positioning of the cantilever and fast restrain the swinging of a hook and the load.
In order to solve the technical problems, the invention provides the following technical scheme: a self-adaptive backstepping nonlinear swing suppression control method of a double-pendulum rotary crane comprises the following steps:
establishing a mathematical model of the double-pendulum rotary crane based on a Lagrange kinetic equation and analyzing characteristics;
establishing an energy function of a mathematical model of the rotary crane according to the characteristics, and establishing an adaptive controller based on the energy function so as to solve the problem of unknown model parameters;
based on a back stepping method and a system control target, establishing a Lyapunov function, and optimizing the self-adaptive controller based on the Lyapunov function; and controlling the swing suppression of the double-pendulum rotary crane through an adaptive controller.
As a preferable scheme of the adaptive backstepping nonlinear swing suppression control method of the double-pendulum rotary crane, the method comprises the following steps: the mathematical model of the rotary crane is expressed as:
order to
G(q)＝[g 1 g z g3g4g5g6]T
N＝[OOOOn5n6]T
F f ＝[f 1 f 2 f 3 f 4 f 5 f 6 ] T
q＝[ζ 1 ζ 2 ζ 3 ζ 4 ζ 5 ζ 6 ] T ；
Wherein: m (q) is the inertia matrix of the double pendulum rotary crane system,
is a centripetal-Coriolis matrix, G (q) is a gravity vector, N is a control input vector, F f Is the mechanical friction force of the double-pendulum rotary crane system, q is the state variable of the double-pendulum rotary crane system,
in the form of the first derivative of the signal,
is its second derivative; m is h And m l Mass of hook and load, M 0 And M 1 Mass of the cantilever and ballast, respectively, /) 1 And l 2 Respectively the length from the suspension rope and the lifting hook to the center of mass of the load, L 1 L is the length of the ballast and the length of the cantilever, I x 、I y 、I z For the moment of inertia of the cantilever in the x, y, z axes, respectively, I b Is the moment of inertia of the ballast, g is the acceleration of gravity, and ζ is a generalized quantity of state describing a rotary crane system i (i 1.., 4) is a swing angle between the hook and the load, and ζ 5 is a cantilever heave direction driving torque, ζ 5 is a driving force/torque 6 For driving torque in the direction of rotation of the cantilever, f 5 、f 6 Mechanical friction in the direction of cantilever heave and rotation, d l And (1., 4) are air friction parameters.
As a preferable scheme of the adaptive backstepping nonlinear swing suppression control method of the double-pendulum rotary crane, the method comprises the following steps: the mathematical model of the rotary crane further comprises,
establishing a friction force feedforward compensation model to eliminate friction force generated by a driving mechanism of the double-pendulum spiral crane, wherein the friction force feedforward compensation model is expressed as follows:
wherein ,f51 、f 52 、f 61 、f 62 、ε 1 and ε2 Feedforward compensation model parameters for friction force, f 51 and f61 Value of (2) and maximum static friction forceCorresponds to f 52 and f62 Is the coefficient of viscous friction, ε 1 and ε2 Is the static coefficient of friction.
As a preferable scheme of the adaptive backstepping nonlinear swing suppression control method of the double-pendulum rotary crane, the method comprises the following steps: the analysis characteristics include analysis of the swing characteristics, load (lever translation characteristics), and the like.
As a preferable scheme of the adaptive backstepping nonlinear swing suppression control method of the double-pendulum rotary crane, the method comprises the following steps: the energy function is expressed as:
wherein ：
respectively representing the cantilever load, hook, swing angle ζ i A speed signal of 1, 4,
is the kinetic energy part of a double-pendulum rotary crane system,
the potential energy portion of its load and hook.
As a preferable scheme of the adaptive backstepping nonlinear swing suppression control method of the double-pendulum rotary crane, the method comprises the following steps: the establishing of the Lyapunov function includes,
the lyapunov equation V is established based on a backstepping method and a system control target according to the dynamic rule of a rotary crane model 1 ：
wherein ：e1 ＝[α 1 α 2 α 3 α 4 α 5 α 6 ] T ，q d ＝[ζ 1d ζ 2d ζ 3d ζ 4d ζ 5d ζ 6d ] T ，e 1 ＝q-q d .e 1 The error of the swing angle and the error of the current position and the target position of the cantilever,
the error of the swing angular speed and the error of the rotation speed of the cantilever are obtained;
for the Lyapunov equation V 1 The derivation is carried out to obtain:
wherein ：k1 Is a positive parameter, e 2 ＝Θ-Θ d ＝[β 1 β 2 β 3 β 4 β 5 β 6 ] T In order to make the derivative of the Lyapunov equation semi-negative, design
Then there are
Design Lyapunov equation V based on backstepping method 2 Expressed as:
wherein ,
to adapt the error values of the column vectors of the system matrix, there are
Is composed of
The error value of the adaptive system vector of (1) is
As a preferable scheme of the adaptive backstepping nonlinear swing suppression control method of the double-pendulum rotary crane, the method comprises the following steps: the adaptive controller is represented as:
lyapunov equation V by design 2 Positive definite, its derivative is semi-negative definite, and can obtain self-adaptive control rate, so that the system can form self-adaptive controller with stable convergence.
As a preferable scheme of the adaptive backstepping nonlinear swing suppression control method of the double-pendulum rotary crane, the method comprises the following steps: the adaptive control rate is expressed as:
as a preferable scheme of the adaptive backstepping nonlinear swing suppression control method of the double-pendulum rotary crane, the method comprises the following steps: the adaptive controller may further comprise a processor for controlling the adaptive controller,
by adjusting k 1 、k 2 Thereby obtaining a gain that quickly adapts to different systems, where k 1 and k2 Is a controller base parameter.
As a preferable scheme of the adaptive backstepping nonlinear swing suppression control method of the double-pendulum rotary crane, the method comprises the following steps: also comprises the following steps of (1) preparing,
the method comprises the following steps of performing tracking control on a double-pendulum rotary crane system by utilizing a reference track of a cantilever, wherein the reference track is a three-section type track:
wherein ,ζd5 ＝30[deg]，ζ d6 ＝45[deg]，ζ d1 ＝15[deg]，ζ d2 ＝30[deg]And q is 5 and 6, the tracking control of 30 degrees is realized in the heave angle within 0 to 5 seconds, the tracking control of 45 degrees is realized in the rotation angle, the tracking control of 15 degrees is realized on the basis of 30 degrees in the heave angle within 5 to 10 seconds, the tracking control of 15 degrees is realized on the basis of 45 degrees in the rotation angle, and the tracking angle is reduced by 30 degrees simultaneously in the last 10 to 15 seconds.
The invention has the beneficial effects that: the design of the self-adaptive control rate has good robustness on system parameters of different rotary cranes; the control method of the PD-like in the self-adaptive controller can realize accurate positioning, and external disturbance is restrained through the feedback of the swing angle information and the design of an error item; the two are combined to realize efficient track tracking and swing suppression of different crane systems, so that a control effect is achieved.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without inventive exercise. Wherein:
fig. 1 is an overall flowchart of an adaptive backstepping nonlinear swing suppression control method for a double-pendulum rotary crane according to a first embodiment of the present invention;
FIG. 2 is a schematic diagram of a mathematical model structure of an adaptive backstepping nonlinear swing suppression control method for a double-pendulum rotary crane according to a first embodiment of the present invention;
FIG. 3 is a schematic diagram of simulation results of an adaptive backstepping nonlinear swing suppression control method for a double-pendulum rotary crane according to a first embodiment of the present invention;
fig. 4 is a schematic diagram of a comparative controller btp (bottom projection planning) simulation result of an adaptive back-stepping nonlinear swing suppression control method for a double-pendulum rotary crane according to a second embodiment of the present invention.
Detailed Description
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, specific embodiments accompanied with figures are described in detail below, and it is apparent that the described embodiments are a part of the embodiments of the present invention, not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without making creative efforts based on the embodiments of the present invention, shall fall within the protection scope of the present invention.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, but the present invention may be practiced in other ways than those specifically described and will be readily apparent to those of ordinary skill in the art without departing from the spirit of the present invention, and therefore the present invention is not limited to the specific embodiments disclosed below.
Furthermore, reference herein to "one embodiment" or "an embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one implementation of the invention. The appearances of the phrase "in one embodiment" in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments mutually exclusive of other embodiments.
The present invention will be described in detail with reference to the drawings, wherein the cross-sectional views illustrating the structure of the device are not enlarged partially in general scale for convenience of illustration, and the drawings are only exemplary and should not be construed as limiting the scope of the present invention. In addition, the three-dimensional dimensions of length, width and depth should be included in the actual fabrication.
Meanwhile, in the description of the present invention, it should be noted that the terms "upper, lower, inner and outer" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the drawings, and are only for convenience of describing the present invention and simplifying the description, but do not indicate or imply that the referred device or element must have a specific orientation, be constructed in a specific orientation and operate, and thus, cannot be construed as limiting the present invention. Furthermore, the terms first, second, or third are used for descriptive purposes only and are not to be construed as indicating or implying relative importance.
The terms "mounted, connected and connected" in the present invention are to be understood broadly, unless otherwise explicitly specified or limited, for example: can be fixedly connected, detachably connected or integrally connected; they may be mechanically, electrically, or directly connected, or indirectly connected through intervening media, or may be interconnected between two elements. The specific meanings of the above terms in the present invention can be understood in specific cases to those skilled in the art.
Example 1
Referring to fig. 1 to 2, an embodiment of the present invention provides a method for controlling adaptive backstepping nonlinear swing suppression of a double-pendulum rotary crane, including:
s1: and establishing a mathematical model of the double-pendulum rotary crane based on a Lagrange kinetic equation and analyzing the characteristics.
Order to
Establishing a mathematical model of the double-pendulum rotary crane with distributed mass load:
G(q)＝[g 1 g z g3g4g5g6lT
N＝[OOOOn5n6]T
F f ＝[f 1 f 2 f 3 f 4 f 5 f 6 ] T
q＝[ζ 1 ζ 2 ζ 3 ζ 4 ζ 5 ζ 6 ] T ；
wherein: m (q) is the inertia matrix of the double pendulum rotary crane system,
is a centripetal-Coriolis matrix, G (q) is a gravity vector, N is a control input vector, F f Is the mechanical friction force of the double-pendulum rotary crane system, q is the state variable of the double-pendulum rotary crane system,
in the form of the first derivative of the signal,
is its second derivative; m is h And m l Mass of hook and load, M 0 And M 1 Mass of the cantilever and ballast, respectively, /) 1 And l 2 Respectively the length from the suspension rope and the lifting hook to the center of mass of the load, L 1 L is the length of the ballast and the length of the cantilever, I x 、I y 、I z For the moment of inertia of the cantilever in the x, y, z axes, respectively, I b Is the moment of inertia of the ballast, g is the acceleration of gravity, and ζ is a generalized quantity of state describing a rotary crane system i (i 1.., 4.) is the swing angle of the hook and the load, and ζ is the driving force/torque 5 For cantilever heave direction drive torque, ζ 6 For driving torque in the direction of rotation of the cantilever, f 5 、f 6 Mechanical friction in the direction of cantilever heave and rotation, d i And (1., 4) are air friction parameters.
Further, a friction force feedforward compensation model is established to eliminate the friction force generated by the driving mechanism of the double-pendulum rotary crane, and the friction force feedforward compensation model is expressed as follows:
wherein ,f51 、f 52 、f 61 、f 62 、ε 1 and ε2 Feedforward compensation model parameters for friction force, f 51 and f61 The value of (A) corresponds to the maximum static friction force, f 52 and f62 Is the coefficient of viscous friction, ε 1 and ε2 Is the static coefficient of friction.
It should be noted that, the friction force feedforward compensation model is used to simply eliminate the friction force effect, so as to effectively avoid the adverse effect of the friction generated by the motion of the driving mechanism on the control effect, for example, the swing caused by the positioning lag/lead is generated to severely increase the difficulty of the anti-swing control.
It should be noted that the inertia matrix of the double pendulum rotary crane system is as follows:
m 13 ＝m l l 1 l 2 (1+ζ 1 ζ 3 )，
m 14 ＝m l l 1 l 2 ζ 1 ζ 4 ，
m 15 ＝(m h +m l )l 1 L(C 5 -ζ 1 S 5 )，
m 16 ＝-(m h +m l )l 1 2 ζ 2 +m l l 1 l 2 ζ 4 ，
m 21 ＝(m h +m l )l 1 2 ζ 1 ζ 2 ，
m 23 ＝m l l 1 l 2 ζ 2 ζ 3 ，
m 24 ＝m 1 l 1 l 2 (1+ζ 2 ζ 4 )，
m 25 ＝-(m h +m l )l 1 Lζ 2 S 5 ，
m 26 ＝(m h +m l )l 1 2 ζ 1 +(m h +m 1 )l 1 LS 5 + m l 1 l 2 ζ 3 ，
m 31 ＝m l l 1 l 2 (1+ζ 1 ζ 3 )，
m 32 ＝m l l 1 l 2 ζ 2 ζ 3 ，
m 35 ＝m l l 2 L(C 5 -ζ 3 S 5 )，
m 41 ＝m l l 1 l 2 ζ 1 ζ 4 ，
m 42 ＝m l l 1 l 2 (1+ζ 2 ζ 4 )，
m 45 ＝-m l l 2 Lζ 4 S 5 ，
m 51 ＝(m h +m l )l 1 L(C 5 -ζ 1 S 5 )，
m 52 ＝-(m h +m l )l 1 Lζ 2 S 5 ，
m 53 ＝m l l 2 L(C 5 -ζ 3 S 5 )，
m 54 ＝-m l l 2 Lζ 4 S 5 ，
m 55 ＝((m h +m l )L 2 +I y )，
m 61 ＝-((m h +ml)l 1 2 ζ 2 +m l l 1 l 2 ζ 4 )，
m 62 ＝(m h +m l )l 1 LS 5 +m l l 1 l 2 ζ 3 +(m h +m l )l 1 2 ζ 1 ，
m 65 ＝-((m h +m l )l 1 Lζ 2 C 5 +m l l 2 Lζ 4 S 5 )，
wherein ,mij The matrix coordinates are represented, i 1, 2 … 6, j 1, 2 … 6.
Centripetal coriolis matrix
The following were used:
c 55 ＝0，
c 61 ＝0，
c 62 ＝0，
c 63 ＝0，
c 64 ＝0，
wherein cij The matrix coordinates are represented, i is 1, 2 … 6, and j is 1, 2 … 6.
Furthermore, the characteristics are analyzed according to a mathematical model of the double-pendulum rotary crane.
It should be noted that the characteristics specifically include a swing characteristic (double pendulum system), a load (rod translation characteristic), and the like thereof, and the energy of the entire system can be analyzed by establishing a mathematical model based on the lagrangian equation and analyzing the characteristics thereof.
S2: and establishing an energy function of a mathematical model of the rotary crane according to the characteristics, establishing a Lyapunov function based on a backstepping method and a system control target, and establishing an adaptive controller based on the energy function and the Lyapunov function.
The energy function of the mathematical model of the double-pendulum tower crane is expressed as follows:
wherein ：
respectively representing the cantilever load, hook, swing angle ζ i A speed signal of 1, 4,
is the kinetic energy part of a double-pendulum tower crane system,
the potential energy portion of its load and hook.
Further, a Lyapunov equation V is established based on a backstepping method and a system control target according to the dynamic rule of a rotary crane model 1 ：
wherein ：e1 ＝[α 1 α 2 α 3 α 4 α 5 α 6 ] T ，q d ＝[ζ 1d ζ 2d ζ 3d ζ 4d ζ 5d ζ 6d ] T ，e 1 ＝q-q d .e 1 The error of the swing angle and the error of the current position and the target position of the cantilever,
the error of the swing angular speed and the error of the rotation speed of the cantilever are obtained;
for the Lyapunov equation V 1 The derivation is carried out to obtain:
wherein ：k1 Is a positive parameter, e 2 ＝Θ-Θ d ＝[β 1 β 2 β 3 β 4 β 5 β 6 ] T In order to make the derivative of the Lyapunov equation semi-negative, design
Then there are
Further, the Lyapunov equation V is designed based on the backstepping method 2 Expressed as:
wherein ,
to adapt the error values of the column vectors of the system matrix, there are
Is composed of
The error value of the adaptive system vector of (1) is
S3: and obtaining the self-adaptive control rate based on the Lyapunov function, and adding the self-adaptive control rate into the self-adaptive controller.
Based on Lyapunov equation V 2 And positive determination and semi-negative determination of a derivative of the positive determination are carried out, so that the self-adaptive control rate can be obtained, and the self-adaptive control rate is added into a self-adaptive controller, so that the system forms a controller with stable convergence, and the real-time regulation and control of the swing inhibition of the double-pendulum rotary crane through the self-adaptive controller are realized.
The adaptive controller is represented as:
wherein the adaptive control rate is expressed as:
further, by adjusting k 1 、k 2 Thereby obtaining a gain of the fast adaptation system, wherein k 1 and k2 Is a controller base parameter.
It should be noted that k in the adaptive controller 1 and k2 Under the condition of giving an initial value, the controller is used for real-time regulation and control, so that better control performance (accurate positioning and quick and effective oscillation elimination) can be obtained in the whole movement process.
In particular, the gain (k) of the PD-like part 1 and k2 ) All gains are positive and can be obtained by trial and error; finally, the correlation parameter f of the feedforward friction model 51 、f 52 、f 61 and f62 After offline recognition, without changing the value of the selection, ε 1 and ε2 Is the static coefficient of friction, which is selected to be 0.01.
Furthermore, the reference track of the fluctuation and rotation of the cantilever is utilized to carry out tracking control on the double-pendulum rotary crane system so as to verify the positioning and pendulum eliminating functions, wherein the reference track is a three-section track represented as:
wherein ,ζd5 ＝30[deg]，ζ d6 ＝45[deg]，ζ d1 ＝15[deg]，ζ d2 ＝30[deg]And q is 5, 6. the tracking control of 30 degrees is realized in 0 to 5 seconds of the heave angle, the tracking control of 45 degrees is realized in the rotation angle, the tracking control of 15 degrees is realized on the basis of 30 degrees in 5 to 10 seconds of the heave angle, the tracking control of 15 degrees is realized on the basis of 45 degrees in the rotation angle, and finally the tracking angle is reduced by 30 degrees in 10 to 15 seconds.
It should be noted that tracking control of the double-pendulum rotary crane system is realized by simulating the pitch angle and the rotation angle of the jib and inputting different reference trajectories for the pitch angle and the rotation angle. The self-adaptive method is verified through simulation software, and the method has good positioning and pendulum-eliminating performance.
Example 2
Referring to fig. 3 to 4, an embodiment of the invention provides a self-adaptive backstepping nonlinear swing suppression control method for a double-pendulum rotary crane, and in order to verify the beneficial effects of the invention, scientific demonstration is performed through simulation experiments.
Compared simulation is carried out by utilizing the controller BTP and the controller using the control method, and the control formula of the controller BTP is as follows:
for a BTP controller, the state vector q ═ θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 ] T Finally, the gain of the controller is k 5p ＝5，k 5d ＝5，k 6p ＝10，k 6d The simulation results obtained using the BTP controller constructed as described above are shown in table 1 below:
table 1: maximum amplitude experiments compare the results.
It can be seen by referring to fig. 2 and 3 that the proposed controller can completely track the target track and realize the positioning function, the BTP controller cannot realize positioning in the rotation and fluctuation directions with positioning errors, the tracking and positioning process of the method is smooth, the positioning task can be completed in 15.50 seconds, for the swing suppression aspect, the amplitude of the hook and the load caused by the controller of the method is not large and cannot exceed 1.95[ deg ], the amplitude of the hook and the load caused by the BTP controller is too large and can reach 2.93[ deg ] at most, and the swing of the method can be completely eliminated within 2-4 seconds after the positioning of the driving mechanism is completed, the suppression effect of the BTP is particularly poor, and the swing is not remained until 20 seconds after multiple violent oscillations, so the method has extremely high swing suppression efficiency, accurate positioning, and no overshoot and steady-state errors.
It should be noted that the above-mentioned embodiments are only for illustrating the technical solutions of the present invention and not for limiting, and although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, which should be covered by the claims of the present invention.
Claims (10)
1. A self-adaptive backstepping nonlinear swing suppression control method of a double-pendulum rotary crane is characterized by comprising the following steps: comprises the steps of (a) preparing a mixture of a plurality of raw materials,
establishing a mathematical model of the double-pendulum rotary crane based on a Lagrange kinetic equation and analyzing characteristics;
establishing an energy function of a mathematical model of the rotary crane according to the characteristics, establishing a Lyapunov function based on a backstepping method and a system control target, and establishing an adaptive controller based on the energy function and the Lyapunov function;
and obtaining the self-adaptive control rate based on the Lyapunov function, and adding the self-adaptive control rate into the self-adaptive controller.
2. The adaptive backstepping nonlinear oscillation suppression control method of the double-pendulum rotary crane according to claim 1, characterized in that: the mathematical model of the double-pendulum rotary crane is expressed as follows:
order to
G(q)＝[g 1 g 2 g 3 g 4 g 5 g 6 ] T
N＝[0 0 0 0 n 5 n 6 ] T
F f ＝[f 1 f 2 f 3 f 4 f 5 f 6 ] T
q＝[ζ 1 ζ 2 ζ 3 ζ 4 ζ 5 ζ 6 ] T ；
Wherein: m (q) is the inertia matrix of the double pendulum rotary crane system,
is a centripetal-Coriolis matrix, G (q) is a gravity vector, N is a control input vector, F f Is the mechanical friction force of the double-pendulum rotary crane system, and q is the state variable of the double-pendulum rotary crane system，
In the form of the first derivative of the signal,
is its second derivative; m is h And m l Mass of hook and load, M 0 And M 1 Mass of the cantilever and ballast, respectively, k 1 And l 2 Respectively the length from the suspension rope and the lifting hook to the center of mass of the load, L 1 L is the length of the ballast and the length of the cantilever, I x 、I y 、I z For the moment of inertia of the cantilever in the x, y, z axes, respectively, I b Is the moment of inertia of the ballast, g is the acceleration of gravity, and ζ is a generalized quantity of state describing a rotary crane system i (i 1.., 4.) is the swing angle of the hook and the load, and ζ is the driving force/torque 5 For cantilever heave direction drive torque, ζ 6 For driving torque in the direction of rotation of the cantilever, f 5 、f 6 Mechanical friction in the direction of cantilever heave and rotation, d l And (1., 4) are air friction parameters.
3. The adaptive backstepping nonlinear oscillation suppression control method of the double-pendulum rotary crane according to claim 2, characterized in that: the mathematical model of the rotary crane further comprises,
establishing a friction feedforward compensation model, wherein the friction feedforward compensation model is expressed as follows:
wherein ,f51 、f 52 、f 61 、f 62 、ε 1 and ε2 Feedforward compensation model parameters for friction force, f 51 and f61 The value of (A) corresponds to the maximum static friction force, f 52 and f62 Is the coefficient of viscous friction, ε 1 and ε2 Is the static coefficient of friction.
4. The adaptive backstepping nonlinear oscillation suppression control method of the double-pendulum rotary crane according to claim 3, characterized in that: the analysis of the characteristics comprises the analysis of the swing characteristics and the load.
5. The adaptive back-stepping nonlinear swing suppression control method of a double-pendulum rotary crane according to claim 1 or 2, characterized in that: the energy function is expressed as:
wherein ：
respectively representing the cantilever load, hook, swing angle ζ i A speed signal of 1, 4,
is the kinetic energy part of a double-pendulum rotary crane system,
the potential energy portion of its load and hook.
6. The adaptive backstepping nonlinear oscillation suppression control method of the double-pendulum rotary crane according to claim 5, characterized in that: the establishing of the Lyapunov function includes,
the lyapunov equation V is established based on a backstepping method and a system control target according to the dynamic rule of a rotary crane model 1 ：
wherein ：e1 ＝[α 1 α 2 α 3 α 4 α 5 α 6 ] T ，q d ＝[ζ 1d ζ 2d ζ 3d ζ 4d ζ 5d ζ 6d ] T ，e 1 ＝q-q d .e 1 The error of the swing angle and the error of the current position and the target position of the cantilever,
the error of the swing angular speed and the error of the rotation speed of the cantilever are obtained;
for the Lyapunov equation V 1 The derivation is carried out to obtain:
wherein ：k1 Is a positive parameter, e 2 ＝Θ-Θ d ＝[β 1 β 2 β 3 β 4 β 5 β 6 ] T In order to make the derivative of the Lyapunov equation semi-negative, design
Then there are
Design Lyapunov equation V based on backstepping method 2 Expressed as:
wherein ,
to adapt the error values of the column vectors of the system matrix, there are
Is composed of
The error value of the adaptive system vector of (1) is
7. The adaptive backstepping nonlinear oscillation suppression control method of the double-pendulum rotary crane according to claim 6, characterized in that: the adaptive controller is represented as:
lyapunov equation V by design 2 And positive determination, wherein the derivative of the positive determination is semi-negative determination, so that a self-adaptive control rate is obtained, and the self-adaptive control rate is added into a self-adaptive controller, so that the system can form a self-adaptive controller with stable convergence.
8. The adaptive backstepping nonlinear oscillation suppression control method of the double-pendulum rotary crane according to claim 7, characterized in that: the adaptive control rate is expressed as:
9. the adaptive backstepping nonlinear oscillation suppression control method of the double-pendulum rotary crane according to claim 7, characterized in that: the adaptive controller may further comprise a processor for controlling the adaptive controller,
by adjusting k 1 、k 2 Thereby obtaining a gain that quickly adapts to different systems, where k 1 and k2 Is a controller base parameter.
10. The adaptive backstepping nonlinear oscillation suppression control method of the double-pendulum rotary crane according to claim 7, characterized in that: also comprises the following steps of (1) preparing,
the method comprises the following steps of performing tracking control on a double-pendulum rotary crane system by utilizing a reference track of a cantilever, wherein the reference track is a three-section type track:
wherein ,ζd5 ＝30[deg]，ζ d6 ＝45[deg]，ζ d1 ＝15[deg]，ζ d2 ＝30[deg]And q is 5, the heave angle realizes 30-degree tracking control in 6.0 to 5 seconds, the rotation angle realizes 45-degree tracking control, the heave angle realizes 15-degree tracking control on the basis of 30 degrees in 5 to 10 seconds, the rotation angle realizes 15-degree tracking control on the basis of 45 degrees, and finally the tracking angle is reduced by 30 degrees in 10 to 15 seconds.
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